A sphere of radius 6cm is melted and recast into spheres of radius 2cm each. How many such spheres can be made?

Options

A. 36
B. 27
C. 24
D. 25

Answer

To solve the problem of how many smaller spheres can be made from a larger sphere that has been melted down, we can follow these steps:

### Step-by-Step Solution:

1. **Calculate the Volume of the Larger Sphere:**
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
For the larger sphere with a radius of 6 cm:
\[ V_{\text{large}} = \frac{4}{3} \pi (6)^3 \]
Calculate \( (6)^3 \):
\[ (6)^3 = 216 \]
Therefore, the volume of the larger sphere is:
\[ V_{\text{large}} = \frac{4}{3} \pi (216) = 288 \pi \text{ cm}^3 \]

2. **Calculate the Volume of the Smaller Sphere:**
Now, we calculate the volume of a smaller sphere with a radius of 2 cm:
\[ V_{\text{small}} = \frac{4}{3} \pi (2)^3 \]
Calculate \( (2)^3 \):
\[ (2)^3 = 8 \]
Therefore, the volume of the smaller sphere is:
\[ V_{\text{small}} = \frac{4}{3} \pi (8) = \frac{32}{3} \pi \text{ cm}^3 \]

3. **Determine the Number of Smaller Spheres:**
To find out how many smaller spheres can be made from the larger sphere, we divide the volume of the larger sphere by the volume of one smaller sphere:
\[ x = \frac{V_{\text{large}}}{V_{\text{small}}} \]
Substituting the volumes we calculated:
\[ x = \frac{288 \pi}{\frac{32}{3} \pi} \]
The \( \pi \) cancels out:
\[ x = \frac{288}{\frac{32}{3}} = 288 \times \frac{3}{32} = \frac{864}{32} = 27 \]

4. **Conclusion:**
Therefore, the number of smaller spheres that can be made is:
\[ \boxed{27} \]